sqrt{3x+12}-1=sqrt{5x+9} eching

$$\sqrt{ 3x+12 } -1= \sqrt{ 5x+9 }$$

x uchun yechish

$$x=-1$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{3x+12}-1\right)^{2}=\left(\sqrt{5x+9}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\sqrt{3x+12}-1\right)^{2} kengaytirilishi uchun ishlating.

$$\left(\sqrt{3x+12}\right)^{2}-2\sqrt{3x+12}+1=\left(\sqrt{5x+9}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{3x+12} ga hisoblang va 3x+12 ni qiymatni oling.

$$3x+12-2\sqrt{3x+12}+1=\left(\sqrt{5x+9}\right)^{2}$$

13 olish uchun 12 va 1'ni qo'shing.

$$3x+13-2\sqrt{3x+12}=\left(\sqrt{5x+9}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{5x+9} ga hisoblang va 5x+9 ni qiymatni oling.

$$3x+13-2\sqrt{3x+12}=5x+9$$

Tenglamaning ikkala tarafidan 3x+13 ni ayirish.

$$-2\sqrt{3x+12}=5x+9-\left(3x+13\right)$$

3x+13 teskarisini topish uchun har birining teskarisini toping.

$$-2\sqrt{3x+12}=5x+9-3x-13$$

2x ni olish uchun 5x va -3x ni birlashtirish.

$$-2\sqrt{3x+12}=2x+9-13$$

-4 olish uchun 9 dan 13 ni ayirish.

$$-2\sqrt{3x+12}=2x-4$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-2\sqrt{3x+12}\right)^{2}=\left(2x-4\right)^{2}$$

\left(-2\sqrt{3x+12}\right)^{2} ni kengaytirish.

$$\left(-2\right)^{2}\left(\sqrt{3x+12}\right)^{2}=\left(2x-4\right)^{2}$$

2 daraja ko‘rsatkichini -2 ga hisoblang va 4 ni qiymatni oling.

$$4\left(\sqrt{3x+12}\right)^{2}=\left(2x-4\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{3x+12} ga hisoblang va 3x+12 ni qiymatni oling.

$$4\left(3x+12\right)=\left(2x-4\right)^{2}$$

4 ga 3x+12 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12x+48=\left(2x-4\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2x-4\right)^{2} kengaytirilishi uchun ishlating.

$$12x+48=4x^{2}-16x+16$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$12x+48-4x^{2}=-16x+16$$

16x ni ikki tarafga qo’shing.

$$12x+48-4x^{2}+16x=16$$

28x ni olish uchun 12x va 16x ni birlashtirish.

$$28x+48-4x^{2}=16$$

Ikkala tarafdan 16 ni ayirish.

$$28x+48-4x^{2}-16=0$$

32 olish uchun 48 dan 16 ni ayirish.

$$28x+32-4x^{2}=0$$

Ikki tarafini 4 ga bo‘ling.

$$7x+8-x^{2}=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-x^{2}+7x+8=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+8 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=7 ab=-8=-8$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -8-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,8 -2,4$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+8=7 -2+4=2$$

Yechim – 7 yigʻindisini beruvchi juftlik.

$$a=8 b=-1$$

-x^{2}+7x+8 ni \left(-x^{2}+8x\right)+\left(-x+8\right) sifatida qaytadan yozish.

$$\left(-x^{2}+8x\right)+\left(-x+8\right)$$

Birinchi guruhda -x ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$-x\left(x-8\right)-\left(x-8\right)$$

Distributiv funktsiyasidan foydalangan holda x-8 umumiy terminini chiqaring.

$$\left(x-8\right)\left(-x-1\right)$$

Tenglamani yechish uchun x-8=0 va -x-1=0 ni yeching.

$$x=8 x=-1$$

\sqrt{3x+12}-1=\sqrt{5x+9} tenglamasida x uchun 8 ni almashtiring.

$$\sqrt{3\times 8+12}-1=\sqrt{5\times 8+9}$$

Qisqartirish. x=8 qiymati bu tenglamani qoniqtirmaydi.

$$5=7$$

\sqrt{3x+12}-1=\sqrt{5x+9} tenglamasida x uchun -1 ni almashtiring.

$$\sqrt{3\left(-1\right)+12}-1=\sqrt{5\left(-1\right)+9}$$

Qisqartirish. x=-1 tenglamani qoniqtiradi.

$$2=2$$

\sqrt{3x+12}-1=\sqrt{5x+9} tenglamasi noyob yechimga ega.

$$x=-1$$